We expect, for physical reasons11.7 that the wave emitted by a time dependent source should
behave like an outgoing wave far from the source. Note that inside the
bounding sphere of the source that need not be true. Earlier in this
chapter, we used an ``outgoing wave Green's function'' to construct the
solution to the IHE with this asymptotic behavior. Well, lo and behold:

(11.107)

For stationary waves (useful in quantum theory)

(11.108)

This extremely important relation forms the connection between free
spherical waves (reviewed above) and the integral equation solutions
we are interested in constructing.

This connection follows from the addition theorems or multipolar expansions of the free spherical waves defined above. For
the special case of these are:

(11.109)

and

(11.110)

From this and the above, the expansion of the Green's functions in
free spherical multipolar waves immediately follows:

(11.111)

and

(11.112)

Note Well: The complex conjugation operation under the sum is
applied to the spherical harmonic (only), not the Hankel
function(s). This is because the only function of the product
is to reconstruct the
via the addition theorem for spherical harmonics. Study this point in
Wyld carefully on your own.

These relations will allow us to expand the Helmholtz Green's
functions exactly like we expanded the Green's function for the
Laplace/Poisson equation. This, in turn, will allow us to precisely
and beautifully reconstruct the multipolar expansion of the vector
potential, and hence the EM fields in the various zones exactly11.8.

This ends our brief mathematical review of free spherical waves and we
return to the description of Radiation.